Universal substitution of formula for propositional variables:

  • Let be formulae.
  • Let be propositional variables. ? The expression denotes the formula obtained by simultaneously replacing all occurences of in by the formula , by and so forth.
  • We say that the formula is a substitution instance of the original formula.
title: Example 1
Let $W$ be $P \rightarrow (Q \rightarrow P)$.
Hence $W(P/\neg P \lor R, Q/\neg P)$ is the formula $\neg P \lor R \rightarrow (\neg P \rightarrow \neg P \lor R)$.
 
In this case, both $W$ and $W(...)$ are both tautologies as $W(...)$ follows from the logic of $W$.
title: Example 2
The formula $(\neg P \lor R \rightarrow \neg P) \rightarrow \neg P \lor R$ is not a subsitution instance of $P \rightarrow (Q \rightarrow P)$.
- Both have a different structure from each other.
If we replace $P$ by $(\neg P \lor R \rightarrow \neg P)$ then we would have to replace $Q$ by $\neg P \lor R$ but we would "miss the consequent of $Q \rightarrow P$".